lsatcvat - Mathbox for Norm Megill

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Theorem lsatcvat 39068

Description: A nonzero subspace less than the sum of two atoms is an atom. (atcvati 32367 analog.) (Contributed by NM, 10-Jan-2015.)

**Hypotheses**
Ref	Expression
lsatcvat.o	⊢ 0 = (0_g‘𝑊)
lsatcvat.s	⊢ 𝑆 = (LSubSp‘𝑊)
lsatcvat.p	⊢ ⊕ = (LSSum‘𝑊)
lsatcvat.a	⊢ 𝐴 = (LSAtoms‘𝑊)
lsatcvat.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lsatcvat.u	⊢ (𝜑 → 𝑈 ∈ 𝑆)
lsatcvat.q	⊢ (𝜑 → 𝑄 ∈ 𝐴)
lsatcvat.r	⊢ (𝜑 → 𝑅 ∈ 𝐴)
lsatcvat.n	⊢ (𝜑 → 𝑈 ≠ { 0 })
lsatcvat.l	⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅))

**Assertion**
Ref	Expression
lsatcvat	⊢ (𝜑 → 𝑈 ∈ 𝐴)

**Proof of Theorem lsatcvat**
Step	Hyp	Ref	Expression
1		lsatcvat.o	. . 3 ⊢ 0 = (0_g‘𝑊)
2		lsatcvat.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑊)
3		lsatcvat.p	. . 3 ⊢ ⊕ = (LSSum‘𝑊)
4		lsatcvat.a	. . 3 ⊢ 𝐴 = (LSAtoms‘𝑊)
5		lsatcvat.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec)
6	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑊 ∈ LVec)
7		lsatcvat.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝑆)
9		lsatcvat.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
10	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑄 ∈ 𝐴)
11		lsatcvat.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
12	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑅 ∈ 𝐴)
13		lsatcvat.n	. . . 4 ⊢ (𝜑 → 𝑈 ≠ { 0 })
14	13	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ≠ { 0 })
15		lsatcvat.l	. . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅))
16	15	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ⊊ (𝑄 ⊕ 𝑅))
17		simpr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → ¬ 𝑄 ⊆ 𝑈)
18	1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 17	lsatcvatlem 39067	. 2 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝐴)
19	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑊 ∈ LVec)
20	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝑆)
21	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑅 ∈ 𝐴)
22	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑄 ∈ 𝐴)
23	13	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ≠ { 0 })
24		lveclmod 21064	. . . . . . . . 9 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
25	5, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod)
26		lmodabl 20866	. . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
27	25, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Abel)
28	2	lsssssubg 20915	. . . . . . . . 9 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
29	25, 28	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊))
30	2, 4, 25, 9	lsatlssel 39015	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
31	29, 30	sseldd 3959	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊))
32	2, 4, 25, 11	lsatlssel 39015	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
33	29, 32	sseldd 3959	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊))
34	3	lsmcom 19839	. . . . . . 7 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
35	27, 31, 33, 34	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
36	35	psseq2d 4071	. . . . 5 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) ↔ 𝑈 ⊊ (𝑅 ⊕ 𝑄)))
37	15, 36	mpbid 232	. . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑅 ⊕ 𝑄))
38	37	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ⊊ (𝑅 ⊕ 𝑄))
39		simpr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → ¬ 𝑅 ⊆ 𝑈)
40	1, 2, 3, 4, 19, 20, 21, 22, 23, 38, 39	lsatcvatlem 39067	. 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝐴)
41	29, 7	sseldd 3959	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
42	3	lsmlub 19645	. . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈))
43	31, 33, 41, 42	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈))
44		ssnpss 4081	. . . . . 6 ⊢ ((𝑄 ⊕ 𝑅) ⊆ 𝑈 → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅))
45	43, 44	biimtrdi 253	. . . . 5 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅)))
46	45	con2d 134	. . . 4 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → ¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈)))
47		ianor 983	. . . 4 ⊢ (¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈))
48	46, 47	imbitrdi 251	. . 3 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈)))
49	15, 48	mpd 15	. 2 ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈))
50	18, 40, 49	mpjaodan 960	1 ⊢ (𝜑 → 𝑈 ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ⊆ wss 3926 ⊊ wpss 3927 {csn 4601 ‘cfv 6531 (class class class)co 7405 0_gc0g 17453 SubGrpcsubg 19103 LSSumclsm 19615 Abelcabl 19762 LModclmod 20817 LSubSpclss 20888 LVecclvec 21060 LSAtomsclsa 38992

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-0g 17455 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-grp 18919 df-minusg 18920 df-sbg 18921 df-subg 19106 df-cntz 19300 df-oppg 19329 df-lsm 19617 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-drng 20691 df-lmod 20819 df-lss 20889 df-lsp 20929 df-lvec 21061 df-lsatoms 38994 df-lcv 39037

This theorem is referenced by: lsatcvat2 39069

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